Determining Calibration Parameters for a Lithographic Process

ABSTRACT

A technique for determining a set of calibration parameters for use in a model of a photo-lithographic process is described. In this calibration technique, images of a test pattern that was produced using the photo-lithographic process are used to determine corresponding sets of calibration parameters. These images are associated with at least three different focal planes in an optical system, such as a photo-lithographic system that implements the photo-lithographic process. Moreover, an interpolation function is determined using the sets of calibration parameters. This interpolation function can be used to determine calibration parameters at an arbitrary focal plane in the photo-lithographic system for use in simulations of the photo-lithographic process, where the set of calibration parameters are used in a set of transmission cross coefficients in the model of the photo-lithographic process.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to systems and techniques for determiningcalibration parameters for a lithographic process.

2. Related Art

Photo-lithography is a widely used technology for producing integratedcircuits. In this technique, a light source illuminates a photo-mask.The resulting spatially varying light pattern is projected onto aphotoresist layer on a semiconductor wafer by an optical system (whichis sometimes referred to as an exposure tool). By developing the3-dimensional pattern produced in this photoresist layer, a layer in theintegrated circuit is created. Furthermore, because there are oftenmultiple layers in a typical integrated circuit, these operations may berepeated using several photo-masks to produce a product wafer.

As dimensions in integrated circuits steadily become a smaller fractionof the wavelength of the light used to expose images of the photo-maskonto the wafer, it is becoming increasingly difficult to design andmanufacture photo-masks that produce the desired target wafer pattern onthe wafer. As a consequence, the structures in or on the idealphoto-mask (also referred to as the target mask pattern) and/or thephysical structures in or on the actual photo-mask bear less and lessresemblance to the desired target wafer pattern. These differencesbetween the photo-mask and the target wafer pattern are used tocompensate for the diffraction and optical-proximity effects that occurwhen light is transmitted through the optics of the exposure tool and isconverted into the 3-dimensional pattern in the photoresist.

When designing a photo-lithographic process (including the target maskpattern(s) for the photo-mask(s) and/or the corresponding sourcepattern(s)), the manufacturing performance of a given solution is oftenevaluated using photo-lithography simulations. In order to improve theaccuracy of the predicted manufacturing performance, a lithographicsimulator that performs these photo-lithography simulations is usuallycalibrated based on a test pattern that is produced using thephoto-lithographic process. However, the calibration is often onlyperformed for a limited range of lithographic conditions in thephoto-lithographic process. Consequently, the simulations may not beaccurate over the full range of lithographic conditions in thephoto-lithographic process, which may adversely impact the predictedmanufacturing performance.

Hence, what is needed is a calibration technique that overcomes theproblems listed above.

SUMMARY OF THE INVENTION

The present disclosure relates to a computer system that determines aset of calibration parameters for use in a model of a photo-lithographicprocess. During operation, the computer system receives images of a testpattern. This test pattern was produced using the photo-lithographicprocess. Furthermore, the images include a first image at a first focalplane in an optical system, a second image at a second focal plane inthe optical system, and a third image at a third focal plane in theoptical system, where the first focal plane, the second focal plane andthe third focal plane are different. Then, the computer systemcalculates a first set of calibration parameters associated with thefirst image, a second set of calibration parameters associated with thesecond image, and a third set of calibration parameters associated withthe third image. Next, the computer system determines an interpolationfunction based on the first set of calibration parameters, the secondset of calibration parameters, and the third set of calibrationparameters, where the interpolation function provides the set ofcalibration parameters corresponding to an arbitrary focal plane in aphoto-lithographic system that implements the photo-lithographicprocess, and where the set of calibration parameters are used in a setof transmission cross coefficients in the model of thephoto-lithographic process.

Note that the interpolation function may include a quadratic function ofa difference between the arbitrary focus plane and a reference focalplane of the photo-lithographic system. This difference may benormalized by a characteristic wavelength used in the photo-lithographicprocess and/or a characteristic wavelength used when capturing theimages. Additionally, the images may include critical-dimensionscanning-electron-microscope images of the test pattern.

Furthermore, the quadratic function may include a constant term whichcorresponds to an average of the first set of calibration parameters,the second set of calibration parameters, and the third set ofcalibration parameters. In addition, the quadratic function may includea linear dependence on the difference.

In some embodiments, one of the first focal plane, the second focalplane and the third focal plane is a focal plane of thephoto-lithographic system. Moreover, calculating a given set ofcalibration parameters (which can be the first set of calibrationparameters, the second set of calibration parameters or the third set ofcalibration parameters) associated with a given focal plane (which canbe the first focal plane, the second focal plane or the third focalplane) may involve comparing a given image (which can be the firstimage, the second image or the third image) with a target imagecorresponding to the test pattern.

In some embodiments, the computer system determines a mask pattern at anobject plane in the photo-lithographic system using an inverse opticalcalculation based on one or more target wafer patterns at one or morefocal planes in the photo-lithographic system and the model of aphoto-lithographic process. This calculation may use one or more sets ofcalibration parameters determined using the interpolation function.

Another embodiment provides a method that includes at least some of theoperations performed by the computer system.

Another embodiment provides a computer-program product for use with thecomputer system. This computer-program product includes instructions forat least some of the operations performed by the computer system.

INCORPORATION BY REFERENCE

All publications, patents, and patent applications mentioned in thisspecification are herein incorporated by reference to the same extent asif each individual publication, patent, or patent application wasspecifically and individually indicated to be incorporated by reference.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features of the invention are set forth with particularity inthe appended claims. A better understanding of the features andadvantages of the present invention will be obtained by reference to thefollowing detailed description that sets forth illustrative embodiments,in which the principles of the invention are utilized, and theaccompanying drawings of which:

FIG. 1 is a drawing illustrating an interpolation function in amulti-dimensional calibration-parameter space for use in a model of aphoto-lithographic process in accordance with an embodiment of thepresent disclosure.

FIG. 2 is a flow chart illustrating a process for determining a set ofcalibration parameters for use in the model of the photo-lithographicprocess in accordance with an embodiment of the present disclosure.

FIG. 3A is a block diagram illustrating an inverse optical calculationin accordance with an embodiment of the present disclosure.

FIG. 3B is a block diagram illustrating a forward optical calculation inaccordance with an embodiment of the present disclosure.

FIG. 4 is a diagram illustrating a mask pattern and correspondinglevel-set functions in accordance with an embodiment of the presentdisclosure.

FIG. 5 is a block diagram illustrating a computer system that performsthe process of FIG. 2 in accordance with an embodiment of the presentdisclosure.

FIG. 6 is a block diagram illustrating a data structure for use in thecomputer system of FIG. 5 in accordance with an embodiment of thepresent disclosure.

FIG. 7 is a block diagram illustrating a data structure for use in thecomputer system of FIG. 5 in accordance with an embodiment of thepresent disclosure.

Note that like reference numerals refer to corresponding partsthroughout the drawings. Moreover, multiple instances of the same partare designated by a common prefix separated from an instance number by adash.

DETAILED DESCRIPTION OF THE INVENTION

The following description is presented to enable any person skilled inthe art to make and use the invention, and is provided in the context ofa particular application and its requirements. Various modifications tothe disclosed embodiments will be readily apparent to those skilled inthe art, and the general principles defined herein may be applied toother embodiments and applications without departing from the spirit andscope of the present invention. Thus, the present invention is notintended to be limited to the embodiments shown, but is to be accordedthe widest scope consistent with the principles and features disclosedherein.

Embodiments of a computer system, a technique for determining a set ofcalibration parameters for use in a model of a photo-lithographicprocess, and a computer program product (i.e., software) for use withthe computer system are described. In this calibration technique, imagesof a test pattern that was produced using the photo-lithographic processare used to determine corresponding sets of calibration parameters.These images are associated with at least three different focal planesin an optical system, such as a photo-lithographic system thatimplements the photo-lithographic process. Moreover, an interpolationfunction is determined using the sets of calibration parameters. Thisinterpolation function can be used to determine calibration parametersat an arbitrary focal plane in the photo-lithographic system for use insimulations of the photo-lithographic process, where the set ofcalibration parameters are used in a set of transmission crosscoefficients in the model of the photo-lithographic process.

By more accurately determining calibration parameters at an arbitraryfocal plane, this calibration technique may facilitate more accuratesimulations of the photo-lithographic process. In turn, thesesimulations may be used to more accurately estimate the manufacturingperformance of a given design of photo-mask(s) and/or source pattern(s)for use in the photo-lithographic process. Consequently, the calibrationtechnique may result in improved designs and shorter design cycles,which may lead to: improved manufacturing yields, shorter time tomarket, and, thus, reducing manufacturing costs.

We now describe embodiments of the calibration technique. In thiscalibration technique, multiple images of a test pattern (such ascritical-dimension scanning-electron-microscope images) are made atdifferent focal planes associated with a photo-lithographic system (suchas an exposure tool) that fabricated the test pattern using aphoto-lithographic process. Using these images, sets of calibrationparameters are determined for at least three different focal planes, forexample, by comparing the images to the test pattern or a target imagethat corresponds to the test pattern. Then, an interpolation function isfit to the sets of calibration parameters. In particular, the argumentof the interpolation function F is the difference between an arbitraryfocal plane (which is sometimes referred to as an image plane) and areference focal plane of the photo-lithographic system (Δf) normalizedto a wavelength λ (such as the characteristic wavelength used in thephoto-lithographic process and/or a characteristic wavelength used whencapturing the images), and the output is the set of calibrationparameters Y at the arbitrary focal plane. Thus, the interpolationfunction may be expressed as

$Y = {{F( \frac{\Delta \; f}{\lambda} )}.}$

Note that the set of calibration parameters Y can then be used insubsequent simulations of the photo-lithographic process, such assimulations of a wafer pattern at the arbitrary focal plane of thephoto-lithographic system based on an illuminated mask pattern at anobject plane of the photo-lithographic system and a model of the opticalpath in the photo-lithographic system.

A wide variety of fitting techniques may be used when determining theinterpolation functions, for example, least-squares fitting ornon-linear fitting techniques (such as a Levenberg-Marquardtleast-squares minimization technique). Moreover, many differentinterpolation functions may be used, such as cubic splines, polynomials,ortho-normal functions, etc. In an exemplary embodiment, theinterpolation function is a quadratic function, i.e.,

${Y = {A_{0} + {A_{1}( \frac{\Delta \; f}{\lambda} )} + {A_{2}( \frac{\Delta \; f}{\lambda} )}^{2}}},$

where A₀ is a constant term which corresponds to an average of at leastthe three sets of calibration parameters, A₁ is the coefficient of thelinear term and A₂ is the coefficient of the quadratic term. Thisquadratic interpolation function is illustrated in FIG. 1, whichpresents a drawing illustrating an interpolation function 110 (with isassociated with a test pattern 112) in a multi-dimensionalcalibration-parameter space for use in a model of the photo-lithographicprocess. In particular, interpolation function 110 fits sets ofcalibration parameters 114 (which are associated with different imagesof test pattern 112) as a function of focal plane 116 to provideestimated calibration parameters 118.

In some embodiments, sets of calibration parameters 114 include atleast: the exposure dose of the scanner in the photo-lithographicsystem, the nominal best focus position in photo-lithographic system,the photo-resist diffusion length or acid diffusion length (which is afunction of the illumination intensity), the photo-mask bias, photo-maskrounding, and/or the illumination source apodization (i.e., how sharp isthe cutoff at the edge of the source pattern). These calibrationparameters may, at least in part, specify the coefficients and themodulation transfer function in a set of transmission cross coefficients(and, in particular, a constant term, a linear term and a quadraticterm) during the simulations (which corresponds to the optical path inthe photo-lithographic system). For example, using estimated calibrationparameters 118 which are inferred from interpolation function 110 at thearbitrary focal plane, the coefficients of kernels in a set oftransmission cross coefficients during the simulations can be expressedin terms of a sub-set of Bessel functions that are ortho-complete overthe simulation domain. Furthermore, the calibration technique may allownon-Kirchhoff three-dimensional photo-mask effects to be included in thesimulations. This approach to determining the interpolation function maybalance the fitting errors and the edge-placement errors in theanalysis. Note that the set of transmission cross coefficients may beexpressed as a large, sparse matrix that may be decomposed and truncatedto extract eigenvalues, as is known to one of skill in the art.

In an exemplary embodiment, the test pattern for a photo-lithographicprocess that produces a 32-nm-node memory device contact layer includesan array of rectangles, such as a 300×130 nm² rectangle that is repeatedon a 400×250 nm² grid within a 6000×6000 nm² expanse. Moreover, theremay be multiple arrays that collectively occupy a large portion of aphoto-mask, e.g., 10×10 mm². In other embodiments, the test patternincludes rectangles, squares and/or hammerhead/dog-bone shapes.

When measuring the images of the test pattern, the scanner wavelengthwas 193 nm. Furthermore, images were measured at 0 and ±30 nm defocus(in other embodiments, ±50 nm defocus is used). The resulting quadraticinterpolation function had A₀ equal to 0.028258, A₁ equal to 0.025525,and A₂ equal to −0.170686. This interpolation function was determinedusing a Levenberg-Marquardt least-squares minimization technique thatminimized the root-mean-square error between a multiple kernel model ofthe transmission cross coefficients and wafer critical-dimensionmeasurements for the test pattern as a function of defocus.

In some embodiments, Levenberg-Marquardt least-squares minimizationtechnique is implemented using a parallel process. During this process,a Jacobian matrix with partial derivatives of simulations of themultiple kernel model of the transmission cross coefficients withrespect to parameters in the multiple kernel model of the transmissioncross coefficients is computed.

Note that the images may be a bitmap or grayscale files that includesets of values corresponding to pixels in the images. In someembodiments, the images include real and imaginary components (orequivalently, magnitude and phase information). For example, images thatinclude magnitude and relative phase information may be measured bygenerating an interference pattern using measurement and reference beamsderived from a common light source or that are spatially and temporallycoherent. Alternatively, phase contrast optics may be utilized.

FIG. 2 presents a flow chart illustrating a process 200 for determininga set of calibration parameters for use in the model of thephoto-lithographic process, which may be performed by a computer system(such as computer system 500 in FIG. 5). During operation, the computersystem receives images of a test pattern (operation 210). This testpattern was produced using the photo-lithographic process. Furthermore,the images include a first image at a first focal plane in an opticalsystem, a second image at a second focal plane in the optical system,and a third image at a third focal plane in the optical system, wherethe first focal plane, the second focal plane and the third focal planeare different. Then, the computer system calculates a first set ofcalibration parameters associated with the first image, a second set ofcalibration parameters associated with the second image, and a third setof calibration parameters associated with the third image (operation212). Next, the computer system determines an interpolation functionbased on the first set of calibration parameters, the second set ofcalibration parameters, and the third set of calibration parameters(operation 214), where the interpolation function provides the set ofcalibration parameters corresponding to an arbitrary focal plane in aphoto-lithographic system (such as an exposure tool) that implements thephoto-lithographic process, and where the set of calibration parametersare used in a set of transmission cross coefficients in the model of thephoto-lithographic process.

In some embodiments, one of the first focal plane, the second focalplane and the third focal plane is a focal plane of thephoto-lithographic system. Moreover, calculating a given set ofcalibration parameters (which can be the first set of calibrationparameters, the second set of calibration parameters or the third set ofcalibration parameters) associated with a given focal plane (which canbe the first focal plane, the second focal plane or the third focalplane) may involve comparing a given image (which can be the firstimage, the second image or the third image) with the target pattern or atarget image corresponding to the test pattern.

In some embodiments, the computer system determines a mask pattern at anobject plane in the photo-lithographic system using an inverse opticalcalculation based on one or more target wafer patterns at one or morefocal planes in the photo-lithographic system and the model of aphoto-lithographic process. This calculation may use one or more sets ofcalibration parameters determined using the interpolation function.

Note that in some embodiments of process 200 there are additional orfewer operations, the order of the operations may be changed, and two ormore operations may be combined into a single operation. For example,more than three images and/or more than three sets of calibrationparameters may be used to determine the interpolation function. In someembodiments, the interpolation function is a closed-form expression.However, in other embodiments the interpolation function is implementedas a look-up table, and values of the calibration parameters for thearbitrary focal plane may be obtained by interpolating between adjacentvalues in the look-up table or by extrapolating one or more valuesproximate to an end of the look-up table.

As noted previously, the manufacturing performance of a particularphoto-lithographic process (including the depth of focus or DOF at agiven exposure latitude or EL, the Mask Error Enhancement Factor orMEEF, etc.) may be evaluated in simulations of the photo-lithographicprocess. Furthermore, mask patterns (corresponding to photo-masks)and/or source patterns may be determined in additional calculations.These simulations and calculations may involve a forward opticalcalculation and/or an inverse optical calculation.

FIG. 3A presents a block diagram illustrating an inverse opticalcalculation 300. In the inverse optical calculation 300, a suitablyilluminated predicted input 310 (such as a mask pattern) is determinedusing an optical path 312 of the photo-lithographic system having anoutput 314 (such as a desired target wafer pattern, for example a layerin a circuit design, which may be represented using a vector-basedformat such as Graphic Design System II and/or OASIS format) in one ofits image planes (which, as noted previously, are also sometimesreferred to as focal planes). In particular,

M=I⁻¹W,

where I is a forward optical path operator (described in FIG. 3B below),I⁻¹ is an inverse optical path operator, W is the target wafer pattern(for example, the layer in the circuit design), and M is the maskpattern. Note that optical path 312 may include illumination and/oroptical effects.

It will be recognized by one of ordinary skill in the art that inverseoptical calculation 300 described in FIG. 3A is ill defined. Inparticular, numerous possible estimated mask patterns may result in thesame desired target wafer pattern. Therefore, the estimated mask patternmay be selected such that it is ‘most likely’ to represent the targetwafer pattern. A variety of constraints and additional criteria may beimposed when determining the solution(s) to this problem in order tofind a unique answer(s). For example, estimated mask patterns that, whenprojected through the optical path of the photo-lithographic system,correspond to the target wafer pattern are more likely to represent theactual photo-mask than other mask patterns (i.e., may have the smallestvalue of an error function that corresponds to the difference of asimulated wafer pattern, which can be determined using a forward opticalcalculation such as that described below with reference to FIG. 3B, andthe target wafer pattern).

The inverse optical calculation 300 may utilize more than one output.For example, one or more target wafer patterns at different focalplanes, different wavelengths and/or different imaging conditions may beused in a mask-pattern simulator. These target wafer patterns mayinclude intensity, magnitude and/or phase information. In someembodiments, the difference of target wafer patterns may be used asoutput 314 in inverse optical calculation 300. Furthermore, in someembodiments each of the target wafer patterns used in inverse opticalcalculation 300 or a term(s) including some combination of the targetwafer patterns may be multiplied by a corresponding weight. In this way,the calculation (and thus, the results) may emphasize one or more of thetarget wafer patterns relative to other target wafer patterns.

In another exemplary embodiment, one or more simulated wafer patternsare determined using a forward optical calculation. This is illustratedin FIG. 3B, which presents a block diagram illustrating a forwardoptical calculation 330. In forward optical calculation 330, a predictedoutput 344 (such as a simulated wafer pattern) is determined using anoptical path 342 having a suitably illuminated input 340 (such as a maskpattern) at one of its object planes. For example, after determining themask pattern using inverse optical calculation 300, its manufacturingperformance (such as the associated process window) may be evaluated bysimulating wafer patterns using forward optical calculation 330 in alithographic simulator. In some embodiments, optical path 342 includessome or all of the aspects of the photo-lithographic process, such as:illumination settings, the electromagnetics of the photo-mask, thestepper optics, etc. In addition, the lithographic simulator may includea model of a photoresist used in the photo-lithographic process, as wellas flare and/or etch effects.

Note that calculations corresponding to one or more optical paths in aninverse optical calculation and/or a forward optical calculation may beimplemented using Fourier-optical techniques. Furthermore, the opticalpaths in an inverse optical calculation and/or a forward opticalcalculation may include multiple models of optical paths, such as forphoto-lithographic processes that involve multiple exposure operationsor exposure tools. Also note that while optical path 312 (FIG. 3A) andoptical path 342 have been traversed in particular directions, each ofthese optical paths may be traversed in either direction.

We now further discuss exemplary embodiments of an inverse opticalcalculation to determine a mask pattern. The inverse optical calculationmay be based on minimization of an error function (which is alsosometimes referred to as a cost function, a merit function or aHamiltonian function). During each iteration of the inverse opticalcalculation, the error function may be a function of the differencebetween a simulated wafer pattern that results when an estimated maskpattern is projected through optical path 342 (FIG. 3B) and the targetwafer pattern. In some embodiments, the estimated mask pattern initiallycorresponds to a target mask pattern, and as the calculation progressesthe estimated mask pattern is allowed to evolve while the target waferpattern is held constant. When multiple target wafer patterns are used(such as target wafer patterns or images at different image planes), insome embodiments the error function (H) equals

${\sum\limits_{j = 1}^{N}{w_{j}{{I_{j} - I_{oj}}}^{n}}},$

where I_(j) is the simulated wafer pattern associated with the forwardprojection of the jth estimated mask pattern (out of N estimated maskpatterns in this example) through optical path 342 (FIG. 3B), w_(j) is acorresponding weight, I_(oj) is the jth target wafer pattern, and n is apower. Note that the error function (H) approaches zero as I_(j)approaches I_(oj).

In an exemplary embodiment, N is 3 and n is 2. The 3 target waferpatterns may be determined at 3 different focal planes (or focussettings) in the photo-lithographic system. For example, with awavelength of 260 nm, the focal planes may be at −600 nm (relative tonominal focus), at 0 nm (i.e., at nominal focus), and 600 nm (relativeto nominal focus). Alternatively or in addition, the 3 target waferpatterns may be determined at three different wavelengths or imagingconditions. Furthermore, a corresponding set of weights {w_(j)} may be1, 0.1, and 1.

In other embodiments, the weights are varied as the inverse opticalcalculation progresses and/or different weights are used for specificparts (or even pixels) of a target wafer pattern (such as differenttopological environments associated with features in the target waferpattern). For example, the weights may be determined based on thedifference between I_(j) and I_(oj) at a given point in the calculation.This approach may exaggerate the features in the circuit design,especially when the calculation is close to a local or global minimumand the error function (H) corresponds to small differences. Thus, ingeneral the error function (H) may be expressed as a double integralover the wafer-pattern area and there may be separate time-dependentweights for I_(j) and I_(oj). Furthermore, in some embodiments the errorfunction (H) is expressed as a relative difference between I_(j) andI_(oj) for at least a portion of the calculation as it progresses.

We now describe an exemplary embodiment of the forward opticalcalculation used when determining a simulated wafer pattern. Forsimplicity, coherent illumination of the mask pattern is utilized.Furthermore, the electric field falling upon the mask pattern isapproximately constant. Thus, the clear regions of the mask pattern passthe light, while the opaque regions block the light. It follows that ascalar electric field E, just behind the mask pattern, may be expressedas

${{E( \overset{\_}{r} )} = \begin{Bmatrix}0 & {chrome} \\1 & {glass}\end{Bmatrix}},$

where {right arrow over (r)}=(x, y) is a point on the (x,y) plane. Thisrepresentation of the mask pattern may be re-expressed using a functionφ (referred to as a level-set function) having positive regions thatindicate glass and negative regions that indicate chrome. Furthermore,the level-set function may equal zero at the boundaries or contours ofthe mask pattern. Therefore, the electric field E associated with themask pattern may be re-expressed as a function of this level-setfunction, i.e.,

E({right arrow over (r)})=ĥ(φ(x,y)),

where ĥ is the Heaviside function

${\hat{h}(x)} = {\begin{Bmatrix}1 & {x \geq 0} \\0 & {x < 0}\end{Bmatrix}.}$

Because an ideal diffraction limited lens acts as a low-pass filter,this may be used as an approximation to the actual (almost but not quiteperfect) lens in the optical path of the exposure tool. Mathematically,the action of the lens may be expressed as

A({right arrow over (r)})=f ⁻¹(Ĉ(f(E({right arrow over (r)}))))

where A({right arrow over (r)}) indicates the electric fielddistribution on the wafer, f indicates the Fourier transform, f⁻¹indicates the inverse Fourier transform, and Ĉ indicates the pupilcutoff function, which is zero for frequencies larger than a thresholddetermined by the numerical aperture of the lens, and one otherwise.Thus, the pupil function is

${{\overset{\Cap}{C}( {k_{x},k_{y}} )} = {{\hat{h}( {k_{\max}^{2} - \lbrack {k_{x}^{2} + k_{y}^{2}} \rbrack} )} = \begin{Bmatrix}0 & {{k_{x}^{2} + k_{y}^{2}} \geq k_{\max}^{2}} \\1 & {{k_{x}^{2} + k_{y}^{2}} < k_{\max}^{2}}\end{Bmatrix}}},$

wherein k_(x), k_(y) and k_(max) represent frequency coordinates inFourier space. Therefore, the aerial image at the wafer is simply thesquare of the electric field

I({right arrow over (r)})=|A({right arrow over (r)})|².

Combining these two equations, we find

F(φ(x,y))=(|f ⁻¹(Ĉ(f(ĥ(φ(x,y)))))|²).

This is a self-contained formula for the image seen by the wafer. Insome embodiments, the simulated wafer pattern is expressed as alevel-set function during the forward optical calculation. Then, whenthe calculation is finished, the simulated wafer pattern may bedetermined by evaluating this level-set function in a plane of the wafer(e.g., the simulated wafer pattern may correspond to values where thelevel-set function equals zero).

Note that this is just one embodiment of the forward projector that canbe used within the scope of this disclosure, chosen by way of exampledue to its relative simplicity. More sophisticated forward opticalcalculations also fall within the scope of the present disclosure. Suchmodels may take into account, by way of example but not limitation,various illumination conditions (e.g., off-axis, incoherent), the actualelectromagnetics of the light field interacting with the photo-mask,various types of photo-masks other than chrome on glass (e.g.,attenuated phase shifting, strong phase shifting, other materials,etc.), the polarization of the light field, the actual properties of thelens (such as aberrations), and/or the vector nature of theelectromagnetic field as it propagates through optical path 342 (FIG.3B).

We now describe the level-set functions in more detail. In the forwardoptical calculation, the mask pattern(s) and/or the simulated waferpattern(s) may be represented using one or more functions having a setof values or a range that is larger than the actual mask pattern(s)and/or the simulated wafer pattern(s). As discussed previously, in oneembodiment the one or more functions include one or more level-setfunctions. This is illustrated in FIG. 4, which presents a mask pattern400 and corresponding level-set functions 414. Pattern 400 includesalternating regions 410, such as glass and chromium (for a mask pattern)or features and spaces (for a simulated wafer pattern). Transitions fromone region to another are characterized by a contour or an edge, such asedge 412. When viewed from a direction perpendicular to a plane of FIG.4, edges (such as edge 412) define pattern 400.

Level-set function 414-1 has two values 416. Edge 412 may correspond toa mid-point between these two values 416. In contrast, level-setfunction 414-2 has three values 418, and edge 412 may correspond tovalue 418-2. While not illustrated in FIG. 4, level-set functions 414extend into the plane of FIG. 4 (i.e., they are 3-dimension functions).As is known to one of skill in the art, there are many alternatelevel-set functions and/or configurations that may be used. For example,in some embodiments one or more separate level-set functions and/orseparate images may be used for the one or more simulated waferpatterns.

As illustrated by level-set function 414-2, in some embodiments thelevel-set function may be expressed as a signed distance functionrelative to the contour or edge 412 (i.e., the value of the level-setfunction in at least a region is a function of the distance from edge412). This formulation may allow effects that occur nearer to edge 412to be highlighted. However, because features in mask patterns andpatterned wafers may occur at random locations (including those farremoved from edge 412), level-set function 414-1 may be useful in thatit provides an equal weighting with respect to edge 412.

In some embodiments, during each iteration of the inverse opticalcalculation the level-set function corresponding to one of the estimatedmask patterns being modified is updated according to

φ_(i+1)=φ_(i) +Δt·∇(H),

where φ_(i+1) is an updated version of the level-set function, φ_(i) isthe current version of the level-set function, Δt is a step size in thecalculation and ∇(H) is a gradient or a derivative of the errorfunction. In an exemplary embodiment, ∇(H) is

${\frac{\delta \; H}{\delta\varphi}_{\phi_{i}}},$

i.e., it is the Frechet derivative of the error function H. Furthermore,in some embodiments ∇(H) is the direction of steepest descent forminimizing or optimizing H by changing φ. Furthermore, in someembodiments a 1^(st) order and/or a 3^(rd) order Runge-Kutta method isused when updating φ_(i). In other embodiments, a Conjugate Gradienttechnique, a Levenberg-Marquardt technique, a Quasi-Newton technique,and/or a Simplex technique may be used.

At least some aspects of Simulated Annealing may be utilized in someembodiments of the inverse optical calculation. In particular, the errorfunction H may be allowed to increase during some steps as thecalculation evolves. In this way, the global minimum in themulti-dimensional space may be determined. Note that the size of thismulti-dimensional space corresponds to a number of quantization levelsto the power of the number of pixels in the estimated mask pattern. Inan exemplary embodiment, these images have at least 1 million pixels(for example, 1024×1024).

In one embodiment, in any iteration of the calculation changes in φ thatdecrease or increase the error function up to 0.5% are performed. If alarger change will result (i.e., ΔH>0.5%), the step size Δt is decreasedby a factor that is at least greater than 1 and the change in φ isimplemented (or not) based on a probability and a value P given by

$^{- \frac{{kH}_{i + 1}}{H_{i}}},$

where H_(i+1) is the error function in the i+1^(th) iteration (if thechange in φ is implemented) and H_(i) is the error function in i^(th)iteration (note that the ratio of H_(i+1)/H_(i) equals 1+ΔH). In someembodiments k is 0.155. For example, if the value P is 0.3 and a randomnumber between 0 and 1 is less than P, the error function is increasedbefore proceeding. In this way, the inverse optical calculationinitially takes large steps and thereby explores the solution space.

Furthermore, in some embodiments, the inverse optical calculation isdivided into a series of overlapping sub-problems (also referred to aswork units) at least some of which are processed independently and/orconcurrently. These work units may be based on elements or structures(for example, repetitive structures) in the target mask pattern(s), thetarget wafer pattern(s), and/or in the simulated wafer pattern(s).Furthermore, in some embodiments the work units may partially overlapneighboring work units. For example, the work units may be between10,000 nm² and 100 μm² in size.

In some embodiments, the inverse optical calculation is run for 100,1000 or 10,000 iterations at which point the optimal solution has beendetermined. In other embodiments, the calculation is stopped based onconvergence criteria, such as: oscillatory behavior, a relative and/orabsolute difference between the simulated wafer pattern associated withthe estimated mask pattern and the target wafer pattern, the latestchange to the error function H, and/or the history of changes to theerror function H. For example, the relative difference may be less than1% and/or the absolute difference may be 10 nm for a CD of 100 nm. Notethat in some embodiments, the level-set function is re-distanced (i.e.,restored to one having the distance function property relative to edge412) at intermediate iterations during the calculation. In an exemplaryembodiment, such re-distancing occurs at least every 20 iterations (forexample, every 14 iterations).

We now discuss computer systems for implementing the calibrationtechnique. FIG. 5 presents a block diagram illustrating a computersystem 500 that performs process 200 (FIG. 2). This computer systemincludes one or more processors 510, a communication interface 512, auser interface 514, and one or more signal lines 522 coupling thesecomponents together. Note that the one or more processors 510 maysupport parallel processing and/or multi-threaded operation, thecommunication interface 512 may have a persistent communicationconnection, and the one or more signal lines 522 may constitute acommunication bus. Moreover, the user interface 514 may include adisplay 516, a keyboard 518, and/or a pointer 520, such as a mouse.

Memory 524 in the computer system 500 may include volatile memory and/ornon-volatile memory. More specifically, memory 524 may include ROM, RAM,EPROM, EEPROM, FLASH, one or more smart cards, one or more magnetic discstorage devices, and/or one or more optical storage devices. Memory 524may store an operating system 526 that includes procedures (or a set ofinstructions) for handling various basic system services for performinghardware dependent tasks. The memory 524 may also store procedures (or aset of instructions) in a communication module 528. The communicationprocedures may be used for communicating with one or more computersand/or servers, including computers and/or servers that are remotelylocated with respect to the computer system 500.

Memory 524 may also include multiple program modules (or sets ofinstructions), including: lithography module 530 (or a set ofinstructions), calibration module 532 (or a set of instructions), anddesign module 534 (or a set of instructions). Note that one or more ofthese program modules (or sets of instructions) may constitute acomputer-program mechanism.

Calibration module 532 may determine one or more interpolation functions536 based on sets of calibration parameters 538 associated with images540 of one or more test patterns 542. These interpolation functions mayallow calibration parameters to be determined at an arbitrary focalplane associated with one or more models of optical paths 544 in aphoto-lithographic system, such as an exposure tool. For example,lithography module 530 may use one of interpolation functions 536 todetermine estimated calibration parameters at the arbitrary focal planewhen simulating wafer patterns 546 based on one or more mask patterns548, one or more source patterns 550 and/or lithographic conditions 552.Note that wafer patterns 546 may be aerial images, or may includedeveloped wafer patterns (such as when lithography module 530 includes aphotoresist model). In some embodiments, lithography module 530 performsmultiple simulations to determine a process window associated with thephoto-lithographic process. This process window may be based on/include:a range of exposure times, a depth of focus, a range of exposureintensities, and/or a normalized image log slope.

Similarly, design module 534 may use one of interpolation functions 536to determine estimated calibration parameters at the arbitrary focalplane when designing one or more mask patterns 548 and/or one or moresource patterns 550 based on one or more target wafer patterns 554.

Instructions in the various modules in the memory 524 may be implementedin a high-level procedural language, an object-oriented programminglanguage, and/or in an assembly or machine language. The programminglanguage may be compiled or interpreted, i.e., configurable orconfigured to be executed by the one or more processors 510.

Although the computer system 500 is illustrated as having a number ofdiscrete items, FIG. 5 is intended to be a functional description of thevarious features that may be present in the computer system 500 ratherthan as a structural schematic of the embodiments described herein. Inpractice, and as recognized by those of ordinary skill in the art, thefunctions of the computer system 500 may be distributed over a largenumber of servers or computers, with various groups of the servers orcomputers performing particular subsets of the functions. In someembodiments, some or all of the functionality of the computer system 500may be implemented in one or more ASICs, one or more field programmablegate arrays (FPGAs), and/or one or more digital signal processors(DSPs).

In some embodiments, the calibration technique may be implemented as astand-alone software application, or as a program module or subroutinein another application, such as photo-lithographic design software.Furthermore, the software may be configured to execute on a client orlocal computer, such as: a personal computer, a laptop computer, orother device capable of manipulating computer readable data, or betweentwo or more computing systems over a network (such as the Internet,World Wide Web or WWW, Intranet, LAN, WAN, MAN, or combination ofnetworks, or other technology enabling communication between computingsystems). Therefore, information used when determining the interpolationfunction(s) may be stored locally (for example, on the local computer)and/or remotely (for example, on a computer or server that is accessedvia a network).

Computer system 500 may include fewer components or additionalcomponents, two or more components may be combined into a singlecomponent, and/or a position of one or more components may be changed.In some embodiments the functionality of the computer system 500 may beimplemented more in hardware and less in software, or less in hardwareand more in software, as is known in the art.

Note that the preceding embodiments may be used for mask patternscorresponding to: chromium-on-glass photo-masks, alternatingphase-shifting photo-masks, attenuating phase-shifting photo-masks,and/or multiple-exposure photo-masks (i.e., where patterns printed usingtwo or more photo-masks are combined to produce a desired pattern).

We now discuss data structures that may be used in the computer system500 (FIG. 5). FIG. 6 presents a block diagram illustrating a datastructure 600. This data structure may include information correspondingto one or more images 610 of a test pattern. For a given image, such asimage 610-1, data structure 600 may include: focal plane 612-1,locations 614-1, pixel values 616-1 associated with locations 614-1, andoptional phases associated with locations 614-1.

FIG. 7 presents a block diagram illustrating a data structure 700. Thisdata structure may include information corresponding to one or moreinterpolation functions 710. For a given interpolation function, such asinterpolation function 710-1, data structure 700 may include:coefficients 712-1, and exponents 714-1 associated with coefficients712-1.

Note that in some embodiments of data structures 600 and/or 700 theremay be fewer or additional components, two or more components may becombined into a single component, and/or a position of one or morecomponents is changed. For example, data structure 600 may includeinformation about the lithographic conditions, such as the type ofillumination (such as disk, point, annulus, sigmas, etc.) and/or thedetails of the optics (such as one or more wavelengths used or thenumerical aperture).

While the preceding discussion has focused on a calibration techniquefor determining an interpolation function for calibration parametersassociated with a photo-lithographic process, in other embodiments thesetechniques may also be applied to determine an interpolation functionfor the calibration parameters associated with a direct write beam inoptical or electron-beam direct-write lithography.

The foregoing descriptions of embodiments of the present invention havebeen presented for purposes of illustration and description only. Theyare not intended to be exhaustive or to limit the present invention tothe forms disclosed. Accordingly, many modifications and variations willbe apparent to practitioners skilled in the art. Additionally, the abovedisclosure is not intended to limit the present invention. The scope ofthe present invention is defined by the appended claims.

1. A computer-implemented method for determining a set of calibrationparameters for use in a model of a photo-lithographic process,comprising: receiving images of a test pattern, wherein the test patternwas produced using the photo-lithographic process, wherein the imagesinclude a first image at a first focal plane in an optical system, asecond image at a second focal plane in the optical system, and a thirdimage at a third focal plane in the optical system, and wherein thefirst focal plane, the second focal plane and the third focal plane aredifferent; calculating, by a computer, a first set of calibrationparameters associated with the first image, a second set of calibrationparameters associated with the second image, and a third set ofcalibration parameters associated with the third image; and determiningan interpolation function based on the first set of calibrationparameters, the second set of calibration parameters, and the third setof calibration parameters, wherein the interpolation function providesthe set of calibration parameters corresponding to an arbitrary focalplane in a photo-lithographic system that implements thephoto-lithographic process; and wherein the set of calibrationparameters are used in a set of transmission cross coefficients in themodel of the photo-lithographic process.
 2. The method of claim 1,wherein the interpolation function includes a quadratic function of adifference between the arbitrary focus plane and a reference focal planeof the photo-lithographic system.
 3. The method of claim 2, wherein thedifference is normalized by a characteristic wavelength used in thephoto-lithographic process.
 4. The method of claim 2, wherein thedifference is normalized by a characteristic wavelength used whencapturing the images.
 5. The method of claim 2, wherein the quadraticfunction includes a constant term which corresponds to an average of thefirst set of calibration parameters, the second set of calibrationparameters, and the third set of calibration parameters.
 6. The methodof claim 2, wherein the quadratic function includes a linear dependenceon the difference.
 7. The method of claim 1, wherein one of the firstfocal plane, the second focal plane and the third focal plane is a focalplane of the photo-lithographic system.
 8. The method of claim 1,wherein calculating a given set of calibration parameters, which can bethe first set of calibration parameters, the second set of calibrationparameters or the third set of calibration parameters, associated with agiven focal plane, which can be the first focal plane, the second focalplane or the third focal plane, involves comparing a given image, whichcan be the first image, the second image or the third image, with atarget image corresponding to the test pattern.
 9. The method of claim1, further comprising determining a mask pattern at an object plane inthe photo-lithographic system using an inverse optical calculation basedon one or more target wafer patterns at one or more focal planes in thephoto-lithographic system and the model of a photo-lithographic process,wherein the determining may involve one or more sets of calibrationparameters determined using the interpolation function.
 10. The methodof claim 1, wherein the images include critical-dimensionscanning-electron-microscope images of the test pattern.
 11. Acomputer-program product for use in conjunction with a computer system,the computer-program product comprising a computer-readable storagemedium and a computer-program mechanism embedded therein for determininga set of calibration parameters for use in a model of aphoto-lithographic process, the computer-program mechanism including:instructions for receiving images of a test pattern, wherein the testpattern was produced using the photo-lithographic process, wherein theimages include a first image at a first focal plane in an opticalsystem, a second image at a second focal plane in the optical system,and a third image at a third focal plane in the optical system, andwherein the first focal plane, the second focal plane and the thirdfocal plane are different; instructions for calculating a first set ofcalibration parameters associated with the first image, a second set ofcalibration parameters associated with the second image, and a third setof calibration parameters associated with the third image; andinstructions for determining an interpolation function based on thefirst set of calibration parameters, the second set of calibrationparameters, and the third set of calibration parameters, wherein theinterpolation function provides the set of calibration parameterscorresponding to an arbitrary focal plane in a photo-lithographic systemthat implements the photo-lithographic process; and wherein the set ofcalibration parameters are used in a set of transmission crosscoefficients in the model of the photo-lithographic process.
 12. Thecomputer-program product of claim 11, wherein the interpolation functionincludes a quadratic function of a difference between the arbitraryfocus plane and a reference focal plane of the photo-lithographicsystem.
 13. The computer-program product of claim 12, wherein thedifference is normalized by a characteristic wavelength used in thephoto-lithographic process.
 14. The computer-program product of claim12, wherein the quadratic function includes a constant term whichcorresponds to an average of the first set of calibration parameters,the second set of calibration parameters, and the third set ofcalibration parameters.
 15. The computer-program product of claim 12,wherein the quadratic function includes a linear dependence on thedifference.
 16. The computer-program product of claim 11, wherein one ofthe first focal plane, the second focal plane and the third focal planeis a focal plane of the photo-lithographic system.
 17. Thecomputer-program product of claim 11, wherein calculating a given set ofcalibration parameters, which can be the first set of calibrationparameters, the second set of calibration parameters or the third set ofcalibration parameters, associated with a given focal plane, which canbe the first focal plane, the second focal plane or the third focalplane, involves comparing a given image, which can be the first image,the second image or the third image, with a target image correspondingto the test pattern.
 18. The computer-program product of claim 11,wherein the computer-program mechanism further includes instructions fordetermining a mask pattern at an object plane in the photo-lithographicsystem using an inverse optical calculation based on one or more targetwafer patterns at one or more focal planes in the photo-lithographicsystem and the model of a photo-lithographic process, wherein thedetermining may involve one or more sets of calibration parametersdetermined using the interpolation function.
 19. The computer-programproduct of claim 11, wherein the images include critical-dimensionscanning-electron-microscope images of the test pattern.
 20. A computersystem, comprising: a processor; a memory; and a program module fordetermining a source pattern to illuminate a photo-mask during aphoto-lithographic process, the program module stored in the memory andconfigured to be executed by the processor to determine a set ofcalibration parameters for use in a model of a photo-lithographicprocess, the program module including: instructions for receiving imagesof a test pattern, wherein the test pattern was produced using thephoto-lithographic process, wherein the images include a first image ata first focal plane in an optical system, a second image at a secondfocal plane in the optical system, and a third image at a third focalplane in the optical system, and wherein the first focal plane, thesecond focal plane and the third focal plane are different; instructionsfor calculating a first set of calibration parameters associated withthe first image, a second set of calibration parameters associated withthe second image, and a third set of calibration parameters associatedwith the third image; and instructions for determining an interpolationfunction based on the first set of calibration parameters, the secondset of calibration parameters, and the third set of calibrationparameters, wherein the interpolation function provides the set ofcalibration parameters corresponding to an arbitrary focal plane in aphoto-lithographic system that implements the photo-lithographicprocess; and wherein the set of calibration parameters are used in a setof transmission cross coefficients in the model of thephoto-lithographic process.